Minimize the objective function $3 x+3 y$ subject to the constraints.
\[
\left\{\begin{array}{l}
2 x+y \geq 12 \\
x+2 y \geq 12 \\
x \geq 0, y \geq 0
\end{array}\right.
\]
The minimum value of the function is $\square$.
(Simplify your answer.)
The value of $x$ is $\square$.
(Simplify your answer.)
The value of $y$ is $\square$.
(Simplify your answer.)
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Answer
The values of \(x\) and \(y\) at which this minimum occurs are both \(\boxed{4}\).
Step 1 ：Define the objective function as \(3x + 3y\).
Step 2 ：The constraints are given by the inequalities \(2x + y \geq 12\), \(x + 2y \geq 12\), and \(x, y \geq 0\).
Step 3 ：Solve the linear programming problem using these constraints.
Step 4 ：The minimum value of the objective function is \(\boxed{24}\).
Step 5 ：The values of \(x\) and \(y\) at which this minimum occurs are both \(\boxed{4}\).
